Aha! Nevermind! I got there.
I was taking the laplace transform of
I wasn't writing the second in terms of . So I did a quick check:
made that change and ended up with
and all suddenly became clear! It then easily works out to the provided answer above...
I hate the frustrating times when studying math, when nothing seems to work or make sense. But then it's so satisfying to figure it out in the end! Sometimes even writing the question out on here helps to shift things about in your mind a bit.